Ocean Engineering. Wave-induced drift of small floating objects in regular waves. Guoxing Huang, Adrian Wing-Keung Law n, Zhenhua Huang

Size: px

Start display at page:

Download "Ocean Engineering. Wave-induced drift of small floating objects in regular waves. Guoxing Huang, Adrian Wing-Keung Law n, Zhenhua Huang"

Grant Johnson
6 years ago
Views:

Ocean Engineering 38 (211) 712 718 Contents lists available at ScienceDirect Ocean Engineering journal homepage: www.elsevier.

Engineering, Nanang Technological Universit, Singapore 639798, Republic of Singapore article info Article histor: Received 19 Jul 21 Accepted 9 December 21 Editor-in-Chief: A.I.

1 Ocean Engineering 38 (211) Contents lists available at ScienceDirect Ocean Engineering journal homepage: Short Communication Wave-induced drift of small floating objects in regular waves Guoing Huang, Adrian Wing-Keung Law n, Zhenhua Huang School of Civil and Environmental Engineering, Nanang Technological Universit, Singapore , Republic of Singapore article info Article histor: Received 19 Jul 21 Accepted 9 December 21 Editor-in-Chief: A.I. Incecik Available online 2 Februar 211 Kewords: Small floating objects Regular waves Wave-induced drift abstract Water waves induce a slow drift of an object floating on the water surface. In this stud, we eamined, b a series of laborator eperiments, the drift motion of small rigid floating objects driven b regular waves in deep water. Different shapes of planar objects, including square, circular and elliptical, were investigated for two different submergences, and their drift motions in waves were determined using an infrared motion monitoring sstem. The corresponding measurements enabled the quantification of the drift characteristics with respect to the wave characteristics and object shapes. Numerical simulations based on an eisting theor were presented and comparisons between the eperimental data and the predictions b the eisting theor were performed. & 21 Elsevier Ltd. All rights reserved. 1. Introduction Water waves induce a slow drift of objects floating on the water surface in the direction of wave propagation. These rigid/fleible floating objects can var vastl in sizes, from small biomass such as phtoplankton, to medium size fleible oil patches (i.e. Kang and Lee, 1995; Law, 1999; Wong and Law, 23), to large rigid floating ice floes in the ocean (Wadhams, 1983). Understanding of their drift behavior is important for engineering purposes. For eample, for floating oil patches in the nearshore region, the wave-induced drift is one of the dominant mechanisms responsible for the beaching of oil patches. A quantitative understanding of the drift behavior is thus necessar for the oil fate and transport modeling (Cheng et al., 2; Law and Huang, 27). For offshore structures in cold regions, drifting icebergs can be etremel hazardous. Generall, wind and ocean currents are considered to be the primar factors causing the iceberg drift (e.g. El-Tahan and El-Tahan, 1983). However, for ice floes in tens-of-meters size range, Wadhams (1983) pointed out that the wave-induced drift can be a dominating factor, even under strong winds. Based on theoretical arguments alone, Arikainen (1972) drew a similar conclusion that for isolated ice floes, the wave-induced drift can be as large as the windgenerated drift and thus should not be neglected. Harms (1987) performed laborator measurements on the drift of ice floe models under regular wave conditions. He obtained an empirical formula to predict the wave-induced drift of these ice floes. Huang (27) studied the variations of the wave-induced surface drift in a wave flume with time and in space. The effects of side-wall on the drift velocit were discussed. n Corresponding author. Tel.: ; fa: address: cwklaw@ntu.edu.sg (A.W.K. Law). In terms of analtical analsis, there are two eisting methods to eamine the motion of a floating object under wave action: one is based on the potential flow theor for large objects and the other is based on Morison s equation for small objects. The first method solves the flow surrounding these large objects which scatter waves, using the surfaces of the objects as flow boundaries. The velocit and pressure fields around the objects can be calculated b the potential theor. As to smaller objects, their disturbance to the wave field can usuall be neglected, and Morison s equation can then be used to predict the wave forces on these objects (e.g. Sorensen, 1978). Both methods had been applied etensivel to compute the wave-induced loadings. However, the have not been well eplored in term of analzing the time-averaged behavior of wave-induced drifting motion for small rigid objects. For small rigid floating objects, Rumer et al. (1979) was probabl the first to use Morison s equation to investigate the wave-induced drift. In their approach, the water surface is considered to be an oscillating slope; the gravit component normal to the slope is balanced b buoanc, and the component tangential to the surface slope induces the movement of the object. Based on the work of Rumer et al. (1979), Shen and Zhong (21) obtained analtical solutions for two special cases where either the added mass coefficient C m or drag coefficient C d vanishes for small amplitude waves in deep-water conditions. Furthermore, the drift velocit of different objects with realistic added mass and drag coefficients were solved numericall. After comparing the results, Shen and Zhong (21) concluded that the wave-induced drift of floating objects will decrease if the added mass coefficient and/or the drag coefficient increase. Theoretical studies on the topic can also be found in Marchenko (1999) and Grotmaack and Melan (26). Grotmaack and Melan (26) compared the models of Rumer et al. (1979) with that of Marchenko (1999). The pointed out that Rumer et al. (1979) incorrectl used the vertical inertia instead of /$ - see front matter & 21 Elsevier Ltd. All rights reserved. doi:1.116/j.oceaneng

2 G. Huang et al. / Ocean Engineering 38 (211) the centripetal force when deriving his equation. Grotmaack and Melan (26) derived a sstem of equations using Hamilton s principle and computed the drift of small objects b a numerical method. The showed that after a sufficientl long time, the floating object either surfs with the wave or moves slowl relative to the wave. This stud investigates eperimentall the wave-induced drift of three-dimensional small rigid floating objects. Different object shapes were investigated for two submergences in order to provide a range of wave-induced inertial force and drag force acting on the objects. The eperimental set-up and data analsis procedure are described in Section 2, and the measurement results are presented in Section 3. The main conclusions are summarized in Section Description of the eperiment 2.1. Eperimental setup and test preparation The measurements were carried out in a wave flume in the Hdraulics Laborator, Nanang Technological Universit, Singapore. The flume was 45 m long, 1.55 m wide and 1.5 m deep. The large flume size allowed deep-water waves to be generated (d/l4 1/2). A piston tpe wave-maker was located at one end of the tank to generate the desired waves. The wave generation sstem was equipped with a DHI Active Wave Absorption Control Sstem (AWACS) to reduce reflection from the wave paddle. At the other end of the wave tank, a wave absorbing beach was used to dissipate the wave energ and reduce the wave reflection. Four capacitance-tpe wave probes were mounted on a steel frame, which was positioned about 6. m from the wave paddle. The wave probes were capable of measuring the water level fluctuations to the nearest.5 mm. The diameter of the probe wire was.6 cm, thus their placement in the water did not cause an significant modification to the wave field. A motion monitoring sstem (Qualiss Track Manager) was installed to capture the trajector of the small moving objects. The sstem consisted of 2 ProRefle infrared cameras, a laptop computer, and several markers (each is 2 cm in diameter and 5 g in weight) coated with reflective material. The two cameras were angled at each other to cover an intersecting span of about 3 m, viewable b both cameras (see Fig. 1). In general, a minimum of three markers on an object were needed to determine the motion of the moving object. When the object moved, however, the rotation of the objects might cause one or two markers to be out of the view of a camera. Therefore, a group of five markers was placed on the small objects in a X pattern to ensure that at least three markers were visible at all time during the eperiments. The images of the markers were continuousl acquired b the two cameras, and the signals were processed to give the instantaneous position of each marker in a calibrated coordinate sstem. The drift behavior of the small objects can be retrieved from the recorded object trajector. In the eperiments, Stokes waves with target periods of 1. and.9 s were eamined. The wave parameters for the drift measurements under deep water conditions are listed in Table 1. The water depth was fied at d¼.8 m, thus the wave lengths were L¼1.56 m for T¼1. s and 1.26 m for T¼.9 s. The wave height H was varied from.2 to.6 m with a.1 m interval. The wave steepness ka (where k is the wave number and a the wave amplitude) thus ranged from.4 to.15. Polethlene plates, with a densit of.96 g cm 3 were used to model the small objects. Two thicknesses, 3. and 4.5 cm, were used to create two different submergences. The models used in the eperiments are shown in Fig. 2. The included the planar shape of Square, Circle and Ellipse-I (planar aspect ratio of 3:4) and Ellipse-II (planar aspect ratio of 1:2). All these shapes were cut out precisel from AutoCAD generated templates. The dimensions of the plates used in the eperiments are listed in Table 2, where L g is the length of the longitudinal ais of the polethlene plates (square, circle or ellipse), L t the length of its transverse ais, and D the uniform thickness of plates. Since it is generall accepted that the effect of wave diffraction is unimportant when L g /Lo.2 and L t /Lo.2 (Isaacson, 1979), the longitudinal length of the plate L g was chosen as.2 m in all our eperiments Eperimental procedures Before starting an eperiment, the desirable water depth d¼.8 m was first established in the wave flume. This was followed b the calibration of the two infrared cameras, where the were placed above the wave flume at a location about 7 m from the wave generator. The water was confirmed to be sufficientl calm b inspecting the motion of the markers for a short period of time. An discernible movement of the markers would suggest the presence of a residual current. Tpicall a lapse of at least 15 min between Table 1 Wave parameters in the eperiments. Ept. series H (m) T (s) d (m) L (m) ka d/l DA DB DC DD DE DF DG DH DI DJ Infrared camera 1 Infrared camera 2 Markers Direction of wave propagation Small object Wave flume ~3. m Fig. 1. Schematic laout of the eperimental set-up.

3 714 G. Huang et al. / Ocean Engineering 38 (211) L t = 2 L t = 2 L g = 2 L g = 2 L t = 15 L t = 1 L g = 2 Wave propagation L g = 2 Fig. 2. Shapes of plates used in the eperiments: (a) square, (b) circle, (c) Ellipse-I and (d) Ellipse-II (dimension in mm). Table 2 Dimensions of the objects used in the eperiments. Shape series Shape codes L g (mm) L t (mm) D (mm) Square Square Square1a Circle Circle Circle1a Ellipse-I Ellipse Ellipse1a Ellipse-II Ellipse Ellipse2a two subsequent runs was required. After it was confirmed that there was no residual current, the wave generator was activated and data recording began. During the tests, the infrared cameras captured the images of the reflective markers at a frequenc of 5 Hz. These images were processed to produce the instantaneous displacement of the object, which were then used to reveal the drift velocit Tpical movement of a polethlene plate Possible effects of wave reflection from the wave absorbent beach were avoided for all runs b limiting the test duration. In the eperiments, the wave phase velocit was 1.56 m/s for T¼1. s and 1.4 m/s for T¼.9 s. The distance from the test span to the end of wave flume was approimate 35 m. Therefore, it would take about 45 s for a wave train to propagate a round trip between the test section and the wave absorbent beach. The eperimental duration was tpicall about 8 s for all runs to monitor the motion behavior of the small objects. As the flume was wider than the one used b Huang (27), the effects of secondar flow on the drift was not significant within this duration. In the drift velocit determination, the data during 2 45 s were used to avoid the reflection effect. Before each test, the polethlene plate was first placed on the still water surface with its longitudinal ais parallel to the direction of wave propagation, which was assessed b bare-ee observations with the sidewall of the wave flume as the reference line (a small tolerance was allowed as it was difficult to make the longitudinal ais strictl parallel to the wave direction). After the waves were generated, the polethlene plate began to drift along the wave flume. Fig. 3 shows several snapshots of the drift of the polethlene plate, Ellipse-II (with L g ¼2 mm, L t ¼1 mm and D¼3 mm), b water waves with a steepness of ka¼.8. Observations showed that the longitudinal ais of the Ellipse-II was nearl parallel to the wave propagating direction at all time; the same had been observed for plates of square shapes as well. Fig. 4 shows the measurements of the horizontal displacement of Square1 under the Ept. series of DG. In the absence of wave reflection, three stages can be identified: (i) from to 5 s, the water was still; (ii) from 5 to 2 s, the object started moving with the waves with a significant acceleration, but a stead state had et to be established; and (iii) from 2 s onward, a quasi-stead state with an approimatel constant drift velocit had been established. This implies that the drift velocit is not a function of the initial position, which was also pointed out b Grotmaack and Melan (26) Determination of the drift velocit The constant drift velocit in the quasi-stead state can be computed b two approaches. One is to obtain the instantaneous mean velocit within one wave period based on an up-crossing method used in analzing irregular waves. In this method, the period-averaged mean drift velocit is a function of time, and can be calculated b dividing the horizontal displacement between two neighboring peaks b the wave period. An eample is shown in Fig. 5, which corresponds to the trajector of the object shown in Fig. 4. Choosing a quasi-stead time interval, from 25 to 45 s, the period-averaged mean velocit calculated from the figure is

G. Huang et al. / Ocean Engineering 38 (211) 712 718 715 t = t = 1 t = 2 t = 3 t = 4 t = 5 Fig. 3. Tpical movement of Ellipse2a tracked b a fied video camera. Waves propagate from left to right.

4 G. Huang et al. / Ocean Engineering 38 (211) t = t = 1 t = 2 t = 3 t = 4 t = 5 Fig. 3. Tpical movement of Ellipse2a tracked b a fied video camera. Waves propagate from left to right. t ¼t/T is the non-dimensional time trace (m) Time (s) Crest drift (m/s) Time (s) Fig. 4. Time histor of horizontal movement recorded b the motion monitoring sstem. Fig. 5. Drift velocit determined b analzing the wave crests..19 m/s. The second method is to calculate the mean drift velocit in the quasi-stead stage b determining the slope of a best-fitting linear trend line from the horizontal trajector. An eample is shown in Fig. 6. After line-fitting the trajector from 25 to 45 s, a drift velocit of.2 m/s is obtained. The drift velocities determined b both approaches are thus almost identical. We shall use the first approach to calculate the drift velocit in this stud. 3. Results and discussion Fig. 7 shows the celerit-normalized drift velocit, u d /c where c is the wave celerit, as a function of the wave steepness, ka. Fig. 7(a) is for the circular and elliptical shapes, and Fig. 7(b) is for the circular and square shapes. Also shown in Fig. 7 is the theoretical Stokes drift. It can be observed that for these small objects, the measured drift

5 716 G. Huang et al. / Ocean Engineering 38 (211) velocities were all significantl larger than Stokes drift. In addition, the circular objects drifted slightl faster than either the square or elliptical objects. The Ellipse-I objects, with an aspect ratio of 3:4 which is close to a circle in terms of the planar shape, drifted in a wa trace (m) Time (s) = R 2 =.979 similar to the square objects, but at a slightl larger velocit compared to the Ellipse-II objects which had an aspect ratio of 1:2. The celerit-normalized drift velocit showed a quadratic dependence on wave steepness ka, which is similar to the Stokes drift. Also, the objects with 45 mm submergence (denoted b suffi a) drifted faster than those with 3 mm submergence regardless of the object shape. The drift behavior can be seen more clearl in Fig. 8, where the Stokes-normalized drift velocit u d /u s is plotted against the wave steepness. The normalized drift decreased when the wave steepness increased, and approached an asmptotic constant value for large wave steepness. Shen and Zhong (21) and Grotmaack and Melan (26) proposed the following equation for the drift motion of a small floating object as follows: ð1þc m Þ ¼ g þ r s A wc d ra b D 9V v9ðv vþ ð1þ Fig. 6. Drift velocit determined b the linear trend line. where v is the instantaneous velocit of the small object, V the wave orbital velocit, g the gravit acceleration, Z the water surface displacement, r s the densit of the small object, r the water densit, A w the wetted area, and A b the bottom surface area of the object. In the following, we performed numerical simulations to Fig. 7. Effect of wave steepness on celerit-normalized drift velocit for different shapes of polethlene plates: (a) circle and Ellipse-I&II, and (b) circle and square.

6 G. Huang et al. / Ocean Engineering 38 (211) Fig. 8. Effect of wave steepness on Stokes-normalized drift velocit for different shapes of polethlene plates: (a) circle and Ellipse-I&II, and (b) circle and square. Table 3 Numerical results of drift velocit based on Eq. (1). D/L A w /A b ka u s /c u d /c (calculated) C d ¼ C m ¼ C d ¼.5 C m ¼ C d ¼.5 C m ¼ calculate the drift velocities of the small objects in the present stud with various added mass and drag coefficients based on Eq. (1). The small objects were all released at the wave crest. Both the surface displacement and the wave orbital velocit were given b the linear wave theor. Sample numerical results are given in Table 3. It can be concluded that: (i) the drift velocit is approimatel equal to the Stokes drift if both the added mass coefficient C m and the drag coefficient C d vanish; (ii) the drift will be reduced when the added mass coefficient and/or drag coefficient increase. It is not difficult to infer that the predicted drift velocit for finite-size objects based on Eq. (1) would alwas be smaller than the Stokes drift as both the added mass coefficient and drag coefficient are not zero in realit. However, our eperimental data showed an opposite trend from the predictions based on the eisting theor. Hence, further research is necessar to clarif the discrepanc. 4. Conclusions In this stud, the wave-induced drift velocit of small floating, thin objects with various shapes and two different submergences were investigated eperimentall under deep water wave conditions. Our results show that the drift velocit of the floating object would increase from rest to reach a quasi-stead constant magnitude within a short time. The constant drift velocit was found to increase with the wave steepness at an approimatel quadratic rate which is similar to the Stokes drift, but the magnitude was higher than the Stokes drift in all cases. These eperimental results differ from the predictions b the eisting theor. We are currentl conducting a theoretical analsis to clarif the discrepanc.

7 718 G. Huang et al. / Ocean Engineering 38 (211) Acknowledgements This work was supported b the Ministr of Education, Singapore, through the AcRF Tier 2 Grant no. MOE28-T2 7. The authors would like to thank the anonmous reviewers for their valuable comments, which improved the qualit of this manuscript. References Arikainen, A.I., Wave drift of an isolated floe. AIDJEX Bulletin No. 16 (Translation of TRUDY Proceedings, AANII, Vol. 33, Leningrad), Division of Marine Resources, Universit of Washington, Seattle, Washington, pp Cheng, N.S., Law, A.W.K., Findikakis, A.N., 2. Oil transport in surf zone. Journal of Hdraulic Engineering, ASCE 126 (11), El-Tahan, M.S., El-Tahan, H.W., Forecast of iceberg ensemble drift. Paper no. OTC-446. In: Proceedings of the 15th Offshore Technical Conference, Houston, Teas. Grotmaack, R., Melan, M.H., 26. Wave forcing of small floating bodies. Journal of Waterwa, Port, Coastal, and Ocean Engineering, ASCE 132 (3), Harms, V.W., Stead wave-drift of modeled ice floes. Journal of Waterwa, Port, Coastal, and Ocean Engineering, ASCE 113 (6), Huang, Z.H., 27. An eperimental stud of the surface drift currents in a wave flume. Ocean Engineering 34, Isaacson, M., Wave-induced forces in the diffraction regime. In: Shaw, T.L. (Ed.), Mechanics of Wave-Induced Forces on Clinders. Pitman Advanced Publishing Program, pp Kang, K.H., Lee, C.M., Stead streaming of viscous surface laer in waves. Journal of Marine Science and Technolog 1, Law, A.W.K., Wave-induced surface drift of an inetensible thin film. Ocean Engineering 26 (11), Law, A.W.K., Huang, G., 27. Observations and measurements of wave-induced drift of surface inetensible film in deep and shallow waters. Ocean Engineering 34 (1), Marchenko, A.V., The floating behavior of a small bod acted upon b a surface wave. Journal of Applied Mathematics and Mechanics 63 (3), Rumer, R.R., Crissman, R., Wake, A., Ice transport in great lakes. Water Resource and Environmental Engineering. Research. Report no State Universit of New York, Buffalo. Shen, H.H., Zhong, Y., 21. Theoretical stud of drift of small rigid floating objects in wave fields. Journal of Waterwa, Port, Coastal, and Ocean Engineering, ASCE 127 (6), Sorensen, R.M., Basic coastal engineering. Wile-Intersciences, New York. Wadhams, P., A mechanism for the formation of ice edge bands. Journal of Geophsical Research 88 (C5), Wong, P.C.Y., Law, A.W.K., 23. Wave-induced drift of an elliptical surface film. Ocean Engineering 3 (3),

PROPAGATION OF LONG-PERIOD WAVES INTO AN ESTUARY THROUGH A NARROW INLET

PROPAGATION OF LONG-PERIOD WAVES INTO AN ESTUARY THROUGH A NARROW INLET Takumi Okabe, Shin-ichi Aoki and Shigeru Kato Department of Civil Engineering Toyohashi University of Technology Toyohashi, Aichi,